Cryptography is a security mechanism for protecting information from unintended disclosure by transforming the information into a form that is unreadable to humans, and unreadable to machines that are not specially adapted to reversing the transformation back to the original information content. The cryptographic transformation can be performed on data that is to be transmitted electronically, such as an electronic mail message, and is equally useful for data that is to be securely stored, such as the account records for customers of a bank or credit company.
In addition to preventing unintended disclosure, cryptography also provides a mechanism for preventing unauthorized alteration of data transmitted or stored in electronic form. After the data has been transformed cryptographically, an unauthorized person is unlikely to be able to determine how to alter the data, because the specific data portion of interest cannot be recognized. Even if the unauthorized user knew the position of the data portion within a data file or message, this position may have been changed by the transformation, preventing the unauthorized person from merely substituting data in place. If an alteration to the transformed data is made by the unauthorized user despite the foregoing difficulties, the fact of the alteration will be readily detectable, so that the data will be considered untrustworthy and not relied upon. This detection occurs when the transformation is reversed; the encrypted date will not reverse to its original contents properly if it has been altered. The same principle prevents unauthorized addition of characters to the data, and deletion of characters from the data, once it has been transformed.
The transformation process performed on the original data is referred to as “encryption.” The process of reversing the transformation, to restore the original data, is referred to as “decryption.” The terms “encipher” and “decipher” are also used to describe these processes, respectively. A mechanism that can both encipher and decipher is referred to as a “cipher.” Data encryption systems are well known in the data processing art. In general, such systems operate by performing an encryption on a plaintext input block, using an encryption key, to produce a ciphertext output block. “Plaintext” refers to the fact that the data is in plain, unencrypted form. “Ciphertext” refers to the fact that the data is in enciphered or encrypted form. The receiver of an encrypted message performs a corresponding decryption operation, using a decryption key, to recover the original plaintext block.
A cipher to be used in a computer system can be implemented in hardware, in software, or in a combination of hardware and software. Hardware chips are available that implement various ciphers. Software algorithms are known in the art as well.
Encryption systems fall into two general categories. Symmetric (or secret key) encryption systems use the same secret key for both encrypting and decrypting messages. An example of a symmetric encryption system is the Data Encryption Standard (DES) system, which is a United States federal standard described in a National Institute of Standards and Technology Federal Information Processing Standard (FIPS Pub 46). In the DES system, a key having 56 independently specified bits is used to convert 64-bit plaintext blocks to 64-bit ciphertext blocks, or vice versa.
Asymmetric (or public key) encryption systems, on the other hand, use two different keys that are not feasibly derivable from one another, one for encryption and another for decryption. A person wishing to receive messages generates a pair of corresponding encryption and decryption keys. The encryption key is made public, while the corresponding decryption key is kept secret. Anyone wishing to communicate with the receiver may encrypt a message using the receiver's public key. Only the receiver may decrypt the message, since only he has the private key. One of the best-known asymmetric encryption systems is the RSA encryption system, named for its originators Rivest, Shamir, and Adleman, and described in U.S. Pat. No. 4,405,829 to Rivest et al., “Cryptographic Communications System and Method.”
A public key system is frequently used to encrypt and transmit secret keys for use with a secret key system. A public key system is also used to provide for digital signatures, in which the sender encrypts a signature message using his private key. Because the signature message can only be decrypted with the sender's public key, the recipient can use the sender's public key to confirm that the signature message originated with the sender.
A commonplace method, for both signature generation and signature verification, is to reduce the message M (to be signed) by means of a cryptographic hash function, in which case, the hash of the message, H(M), is signed instead of the message M itself. Signing H(M) requires only one encryption operation whereas signing M may require several encryption operations, depending on the length of M.
One of the serious concerns regarding most public key cryptography systems is that, since some public keys can be weaker (although very few, some are easier to “break” than others), an ill intentioned party may deliberately generate such a weaker key to be used to encrypt certain valuable information or to electronically sign a document. This party, if it finds it useful for its purposes, may claim that since the “weak” key was used for a particular transaction, an attacker could decrypt the message or forge a signature. This cheating user can then require the annulment of the transaction or other actions based on such a weakness in the key. For example, if a key used in the Elliptic Curve cryptography system is such that the private key, usually denoted as d, is short, then the encryption can be solved much faster than in the general case. A danger is becoming even more apparent when one is dealing with cryptography systems based on a popular and widely accepted RSA algorithm.
Such a danger exists only when the cheating party (often referred to as “the first party” in the situations described above since the party is responsible for the generation of the key which constitutes the first step in the encryption or in a signing protocol) puts a significant effort into generating such “bad” keys since the probability of obtaining one at random is extremely small. The purpose of this invention is to develop a protocol which will ultimately stop these efforts of a first party attacker and thus assure all parties in the quality of the public keys and in the non-repudiation of the signatures generated.
The approach herein is described with respect to the RSA signature algorithm although it is equally applicable to use in other public key cryptography systems requiring the generation of primes.
A method for computing digital signatures with the RSA algorithm is described in ANSI Standard X9.31-1998 Digital Signatures Using Reversible Public Key Cryptography For The Financial Services Industry (rDSA). ANSI Standard X9.31 defines procedures for:                i. Choosing the public verification exponent, e,        ii. Generating the private prime factors, p and q, and public modulus, n=pq, and        iii. Calculating the private signature exponent, d.        
The procedure for signing a message M (signature production) consists of the following steps: M is hashed using a cryptographic hash function H to produce a hash value H(M). H(M) is then encapsulated within a data structure IR, a representative element RR is computed from IR, and RR is raised to the power d modulo n. The signature Σ is either the result or its complement to n, whichever is smaller. That is, Σ=min{RRd mod n, n-(RRd mod n)}. The signature Σ is exactly one bit less in length than the length of the modulus n. The message and signature (M, Σ) are then sent to the receiver for verification.
The procedure for verifying a signature (signature verification) consists of the following steps: The verifier treats the message and signature as (M′, Σ′) until the signature verification is successful, and it is proven that M=M′ and Σ=Σ′. The signature Σ′ is raised to the power e mode n in order to obtain the intermediate integer RR′. That is, RR′=(Σ′)e mod n. The intermediate integer IR′ is then computed from RR′ as a function of the least significant (right most) bits of RR′. A sanity check is then performed on IR′, and if this step succeeds, the value of H(M)′ is then recovered from IR′. Finally, a hash is computed on the received message M′, and the computed value of H(M′) is compared for equality with the recovered value of H(M)′. The verification process succeeds if the two values are equal and it fails if the two values are not equal.
The public exponent is a positive integer e, where 2≦e≦2k−160, and k is the length of the modulus n in bits. The public exponent may be selected as a fixed value or generated as a random value. When e is odd, the digital signature algorithm is called RSA. When e is even, the digital signature algorithm is called Rabin-Williams. Common fixed values for e are 2, 3, 17, and 216+1=65,537.
The public modulus, n, is the product of two distinct positive primes, p and q (i.e., n=pq).
The private prime factors, p and q, are secretly and randomly selected by each signing entity. The private prime factors must satisfy several conditions, as follows:    1. Constraints on p and q relative toe are:            If e is odd, then e shall be relatively prime to both p−1 and q−1.        If e is even, then p shall be congruent to 3 mod 8, q shall be congruent to 7 mod 8, and e shall be relatively prime to both (p−1)/2 and (q-1)/2.            2. The numbers p±1 and q±1 shall have large prime factors greater than 2100 and less then 2120, such that:            p−1 has a large prime factor denoted by p1         p+1 has a large prime factor denoted by p2         q−1 has a large prime factor denoted by q1         q+1 has a large prime factor denoted by q2             3. The private prime factor p is the first discovered prime greater than a random number Xp, where (√{square root over ( )}2) (2511+128s)≦Xp≦(2512+128s−1), and meets the criteria in Nos. 1 and 2 above, and the private prime factor q is the first discovered prime greater than a random number Xq, where (√{square root over ( )}2) (2511+128s)≦Xq≦(2512+128s−1), and meets the criteria in Nos. 1 and 2 above. s=0, 1, 2, etc. is an integer used to fix the block size. Once selected, the value of s remains constant for the duration of the prime generation procedure.    4. The random numbers Xp and Xq must be different by at least one of their first most significant 100 bits, i.e., |Xp−Xq|>2412+128s. For example, if s=4, so that Xp and Xq are 1024-bit random numbers, then the most significant bit of Xp and the most significant bit of Xq must be “1” and the next most significant 99 bits of Xp and the next most significant 99 bits of Xq must be different in at least 1 bit. Likewise, the private prime factors, p and q, must also satisfy the relationship |p−q|>2412+128s.
The private signature exponent, d, is a positive integer such that d>2512+128s. That is, the length of d must be at least half the length of the modulus n. d is calculated as follows:                If e is odd, then d=e−1 mod (LCM (p−1, q−1))        If e is even, then d=e−1 mod (½ LCM (p−1, q−1))where LCM denotes “Least Common Multiple.” In the rare event that d≦2512+128s, then the key generation process is repeated with new seeds for Xq1, Xq2, and Xq. The random numbers Xq1 and Xq2 are defined below.        
The candidates for the private prime factors, p and q, are constructed using the large prime factors, p1, p2, q1, and q2, and the Chinese Remainder Theorem (see A. Menezes, P. C. Van Oorschot, and S. Vanstone, Handbook of Applied Cryptography, CRC Press, 1997.)
The large prime factors p1, p2, q1, and q2, are generated from four generated random numbers Xp1, Xp2, Xq1 and Xq2. The random numbers are chosen from an interval [2100+a, 2101+a−1] where “a” satisfies 0≦a≦20. For example, if a=19, then the random numbers are randomly selected from the interval [2119, 2120−1], in which case the random numbers each have 120 bits, where the most significant bit of each generated number Xp1, Xp2, Xq1 and Xq2 is “1.” If a pseudo random number generator (PRNG) is used, it is recommended that the four random numbers should be generated from 4 separate input seeds.
The p1, p2, q1, and q2, are the first primes greater than their respective random X values (Xp1, Xp2, Xq1 and Xq2) and such that they are mutually prime-with the public exponent e. That is, e must not contain a factor equal to p1, p2, q1, or q2.
The procedure to generate the private prime factor p is as follows:
A. Select a value of s, e.g., s=4.
B. Generate a random number Xp such that (√{square root over ( )}2) (2512+128s)≦Xp≦(2512+128s−1).
C. Compute the intermediate values:
                Rp=(p2−1 mod p1)p2−(p1−1 mod p2)p1. If Rp<0, then replace Rp by        Rp+p1p2.        Y0=Xp+(Rp−Xp mod p1p2).        
If e is odd, do the following:
1. If Y0<Xp, replace Y0 by (Y0+p1p2). Y0 is the least positive integer greater than Xp congruent to (1 mod p1) and (−1 mod p2). This ensures that p1 is a large prime factor of (Y0−1) and p2 is a large prime factor of (Y0+1).
2. Search the integer sequence
{Y0, Y1=Y0+(p1p2), Y2=Y0+2(p1p2), Y3=Y0+3(p1p2), . . . , Y1=Y0+I(p1p2)}
in order, where I is an integer ≧0, until finding a Yi such that
                Yi is prime and        GCD (Yi−1, e)=1where GCD is Greatest Common Divisor, in which case, p=Yi.        
If e is even, do the following:
1. Replace Y0 with Y0+kp1p2, where 0≦k≦7 is the smallest non-negative integer that makes Y0+kp1p2=3 mod 8. If Y0<Xp, replace Y0 by (Y0+8p1p2). Y0 is the least positive integer greater than Xp congruent to (1 mod p1) and (−1 mod p2) and (3 mod 8).
2. Search the integer sequence.
{Y0, Y1=Y0+(8p1p2), Y2=Y0+2(8p1p2), Y3=Y0+3(8p2p2), . . . , Yi=Y0+I(8p1p2)}
in order, where I is an integer ≧0, until finding a Yi such that
                Yi is prime and        GCD ((Yi−1)/2, e)=1, and Yi=3 mod 8in which case, p=Yi.        
The procedure to generate the private prime factor q is as follows:
A. The s used to generate q is the same s used to generate p, e.g., s=4.
B. Generate a random number Xq such that (√{square root over ( )}2) (2511+128s)≦Xq≦(2512+128s−1).
C. Compute the intermediate values:
                Rq=(q2−1 mod q1) q2−(q1−1 mod q2) q1. If Rq<0, then replace Rq by        Rq+q1q2.        Y0=Xq+(Rq−Xq mod q1q2).        
If e is odd, do the following:
1. If Y0<Xq, replace Y0 by (Y0+q1q2). Y0 is the least positive integer greater than Xq congruent to (1 mod q1) and (−1 mod q2). This ensures that q1 is a large prime factor of (Y0−1) and q2 is a large prime factor of (Y0+1).
2. Search the integer sequence
{Y0, Y1=Y0+(q1q2), Y2=Y0+2(q1q2), Y3=Y0+3(q1q2), . . . , YiY0+I(q1q2)}
in order, where I is an integer ≧0, until finding a Yi such that
                Yi is prime and        GCD (Yi−1, e)=1in which case, q=Yi.        
If e is even, do the following:
1. Replace Y0 with Y0+kq1q2, where 0≦k≦7 is the smallest non-negative integer that makes Y0+kq1q2=7 mod 8. If Y0<Xp, replace Y0 by (Y0+8q1q2). Y0 is the least positive integer greater than Xq congruent to (1 mod q1) and (−1 mod q2) and (7 mod 8).
2. Search the integer sequence.
{Y0, Y1=Y0+(8q1q2), Y2=Y0+2(8q1q2), Y3=Y0+3(8q1q2), . . . , Yi=Y0+I(8q1q2)}
in order, where I is an integer ≧0, until finding a Yi such that
                Yi is prime and        GCD ((Yi−1)/2, e)=1, and Yi=7 mod 8 in which case, q=Yi.        
As mentioned above, the value |Xp−Xq| must be >2412+128s. If not, then another Xq is generated, and a new value of q is computed. This step is repeated until the constraint is satisfied. Likewise, the generated values of p and q must satisfy the relation |p−q|>2412+128s.
NOTE: It is very unlikely that the test on |Xp−Xq| would succeed and the test on |p−q| would fail.
NOTE: According to the X9.31 Standard, if a pseudo random number generator is used to generate random numbers, then separate seeds should be used to generate Xp and Xq.
Altogether there are six random numbers Xi needed in the generation of the private prime factors, p and q, namely Xp, Xp1, Xp2, Xq, Xq1 and Xq2. These random numbers are generated by either a true noise hardware randomizer (RNG) or via a pseudo random generator (PRNG).
The random numbers Xi are generated differently depending on whether or not the process of generating the private prime factors (p and q) requires the capability to be audited later by an independent third party.
In the case where no audit is required, the outputs of the RNG and the PRNG are used directly as the random numbers Xi. In the case where audit is required, the outputs of the RNG and the PRNG are used as intermediate values, called SEED (upper case) values, and these SEED values are then hashed to produce the random numbers Xi. That is, when an audit capability is required, an extra hashing step is used in the generation of the random numbers Xi.
The PRNG itself makes use of an input seed (lower case), which is different from the generated SEED values. Thus, when an audit capability is required and a PRNG is used, a random seed (lower case) is input to the PRNG and a SEED (upper case) is output from the PRNG.
To illustrate the process, suppose that one wishes to generate a 1024 bit random number Xi using SHA-1 as the hash algorithm—see ANSI Standard X9.30-1996, Public Key Cryptography Using Irreversible Algorithms for the Financial Services Industry, Part 2: The Secure Hash Algorithm—1 (SHA-1). Since the output hash value from SHA-1 is 160 bits, the optimal method for generating an Xi of 1024 bits is to generate 7 160-bit SEED values, denoted SEED1 through SEED7, hash each of these SEED values with SHA-1 to produce 7 corresponding 160-bit hash values, denoted hash1 through hash7, and then extract 1024 bits from the available 1120 bits, e.g., by concatenating the values hash1 through hash6 together with 64 bits taken from hash7.
The method for generating a 120-bit Xi is more straightforward. In this case, a single 160-bit SEED is generated and then hashed, and 120 bits are taken from the resulting hash value. The concatenation of the 7 SEED values used in generating each of Xp and Xq are denoted XpSEED and XqSEED, respectively. The single SEED values used in generating Xp1, Xp2, Xq1 and Xq2 are denoted Xp1SEED, Xp2SEED, Xq1SEED, and Xq2SEED, respectively.
In order to allow for audit, the SEED values XpSEED, XqSEED, Xp1SEED, Xp2SEED, Xq1SEED, and Xq2SEED must be saved, and they must be available in case an audit is required. The SEED values must also be kept secret. It is recommended that the SEED values (XpSEED, XqSEED, Xp1SEED, Xp2SEED, Xq1SEED, and Xq2SEED) be retained with the private key as evidence that the primes were generated in an arbitrary manner.
The procedure for auditing the generation procedure (i.e., the generation of the private prime factors,p and q) is a follows:    1. The inputs to the audit procedure are the public exponent e, the public modulus n, and the six secret SEED values XpSEED, XqSEED, Xp1SEED, Xp2SEED, Xq1SEED, and Xq2SEED.    2. The SEED values XpSEED, XqSEED, Xp1SEED, Xp2SEED, Xq1SEED, and Xq2SEED are hashed, and the random number Xp, Xq, Xp1, Xp2, Xq1 and Xq2, are produced from the generated hash values, respectively, using the same procedure that was used to generate the private prime factors, p and q.    3. The private prime factors, p and q, and the private signature exponent d are re-generated using the same procedure used originally to generate p, q, and d.    4. The generated p and q are multiplied together and the resulting product is compared for equality with the input modulus n. If the two values are equal, then the prime factors were generated according to the rules prescribed in the ANSI X9.31 Standard. Otherwise, the prime factors were not generated according to the rules prescribed in the ANSI X9.31 Standard.
The audit procedure is specifically designed to defend against a so-called First Party Attack. In a first party attack, a user purposely generates a large number of candidate prime numbers until one is found that has some mathematical weakness. Later, the user repudiates one or more of his generated signatures by showing that a weakness exists in one of his primes and claiming that the weakness was discovered and exploited by an adversary. In such a case, the user (or First Party) does not follow the prescribed ANSI X9.31 private prime factor generation procedure, but instead uses a different method to purposely construct primes that have a desired weakness. Even if one generates a pair of strong primes as required by the ANSI X9.31 standard, it is still possible that the primes are “bad”, that is, that the primes have such undesirable properties as p/q is near the ratio of two small integers or |p−q| does not have a large prime factor or that GCD(p−1, q−1) is small. While these situations are highly unlikely to occur, it is possible that a persistent attacker may find such primes and use them to deliberately generate a “bad” public key.
But, the ANSI X9.31 method of prime number generation—hashing SEED values to generate the needed random numbers, Xi—prevents an insider from starting with an intentionally constructed “bad prime” and working backwards' to derive the SEED(s) needed to generate the prime. Whereas, it might be possible to start with a constructed “bad prime” and invert the steps to obtain the corresponding random number Xi (needed to produce the “bad prime”), it is not possible to invert the hash function to determine the required input SEED(s) that will produce Xi. In effect, the method of using hash values forces the user to generate his primes using a “forward process.” This means that the only way a “bad prime” can be produced is by pure chance—by repeatedly selecting different starting SEED values and generating primes from these SEED values until a “bad prime” happens to be produced. However, the probability of such a chance event is very small, in fact small enough so that (for practical purposes) a user will never be able to find a “bad prime” using trial and error.
The procedure for generating the private prime factors (p and q) and the private signature exponent d can be specified in terms of the following abbreviated steps:    1. Generate Xp1SEED, and then generate Xp1 from Xp1 SEED. This is a constructive step that cannot fail.    2. Generate p1 from Xp1. This step is an iterative step in which candidate values of p1 are generated from a single starting value Xp1, in a prescribed order, until a p1 is found that satisfies a required primality test and a test involving the public verification exponent e. The step can potentially fail in the very unlikely event that the size of the generated p1 (e.g., 121 bits) is greater than the size of the starting value Xp1 (e.g., 120 bits). If step 2 fails, then repeat steps 1 and 2; otherwise, continue with step 3.    3. Generate Xp2SEED, and then generate Xp2 from Xp2SEED. This is a constructive step that cannot fail.    4. Generate p2 from Xp2. This step is an iterative step in which candidate values of p2 are generated from a single starting value Xp2, in a prescribed order, until a p2 is found that satisfies a required primality test and a test involving the public verification exponent e. The step can potentially fail in the very unlikely event that the size of the generated p2 (e.g., 121 bits) is greater than the size of the starting value Xp2 (e.g., 120 bits). If step 4 fails, then repeat steps 3 and 4; otherwise, continue with step 5.    5. Generate XpSEED (e.g., consisting of 7 160-bit SEEDs), and then generate Xp from XpSEED. This step involves a test to ensure that Xp falls within a specified range of allowed values. The step is repeated (possibly several times) until a suitable value of Xp is found.    6. Generate p from Xp. This step is an iterative step in which candidate values of p are generated from a single starting value Xp, in a prescribed order, until a p is found that satisfies a required primality test and a test involving the public verification exponent e. The step can potentially fail in the extremely unlikely event that the size of the generated p (e.g., 1025 bits) is greater than the size of the starting value Xp (e.g., 1024 bits). If step 6 fails, then repeat steps 5 and 6; otherwise, continue with step 7    7. Generate Xq1SEED, and then generate Xq1 from Xq1 SEED. This is a constructive step that cannot fail.    8. Generate q1 from Xq1. This step is an iterative step in which candidate values of q1 are generated from a single starting value Xq1 in a prescribed order, until a q1 is found that satisfies a required primality test and a test involving the public verification exponent e. The step can potentially fail in the very unlikely event that the size of the generated q1 (e.g., 121 bits) is greater than the size of the starting value Xq1 (e.g., 120 bits). If step 8 fails, then repeat steps 7 and 8; otherwise, continue with step 9.    9. Generate Xq2SEED, and then generate Xq2 from Xq2SEED. This is a constructive step that cannot fail.    10. Generate q2 from Xq2. This step is an iterative step in which candidate values of q2 are generated from a single starting value Xq2, in a prescribed order, until a q2 is found that satisfies a required primality test and a test involving the public verification exponent e. The step can potentially fail in the very unlikely event that the size of the generated q2 (e.g., 121 bits) is greater than the size of the starting value Xq2 (e.g., 120 bits). If step 10 fails, then repeat steps 9 and 10; otherwise, continue with step 11.    11. Generate XqSEED (e.g., consisting of 7 160-bit SEEDs), and then generate Xq from XqSEED. This step involves a test to ensure that Xq falls within a specified range of allowed values and that |Xp−Xq| is greater than a specified value. The step is repeated (possibly several times) until a suitable value of Xq is found.    12.Generate q from Xq. This step is an iterative step in which candidate values of q are generated from a single starting value Xq, in a prescribed order, until a q is found that satisfies a required primality test and a test involving the public verification exponent e and a test to ensure that |p−q| is greater than a specified value. The step can potentially fail in the extremely unlikely event that the size of the generated q (e.g., 1025 bits) is greater than the size of the starting value Xq (e.g., 1024 bits). If step 12 fails, then repeat steps 11 and 12; otherwise, continue with step 13.    13.Generate the private signature exponent d from e, p and q. Then test d to ensure that it is smaller than a specified value. In the extremely rare event that the test on d fails, repeat steps 7 through 13; otherwise stop.
The ANSI X9.31 prescribed audit procedure has certain disadvantages.    1. For a modulus with 1024-bit primes, approximately 2880 bits of extra SEED values (XpSEED, XqSEED, Xp1SEED, Xp2SEED, Xq1SEED, and Xq2SEED) would need to be carried with each private key. This more than triples the number of secret bits that need to be carried in the “private key.”    2. Although the ANSI X9.3 1 Standard recommends that the SEED values be retained with the private key, some implementers may object to this (e.g., when the key is stored on a smart card or when the key is input to a cryptographic function or cryptographic hardware to perform a cryptographic operation), and they may elect to retain the SEED values separately from the private key. But keeping the SEED values separate from the private key has even worse ramifications. In that case, the SEED values may become lost or damaged, in which case the audit function is crippled or rendered ineffective, and most likely the signature key is also rendered ineffective. The user must also protect the secrecy of the of the SEED values, since if the SEED values are discovered by an adversary, they can be easily used to generate the primes and private key, and hence to forge signatures. Storing the private key and SEED values in two different locations means that there are now two targets or points of attack for the adversary, not just one. Thus, when the audit feature is used, the honest user must take extra special steps to prevent the SEEDs from becoming lost or damaged and to protect the secrecy of the SEEDs. This places an extra burden on the user.    3. The SEED values are independent, and there is no apparent need for this. Consequently, it might be possible for an insider attacker to exploit this degree of independence to attack the procedure.
As noted above, though, it is possible to generate “bad primes” through chance or through repeated generation and testing. A significant problem with public key encryption is that some of the public keys might be weaker than others and therefore, if used, will give an attacker a greater chance of breaking the encryption and discovering the secret. For example, a prime number p used in the RSA encryption algorithm would be considered weak for the purposes of RSA encryption or signature generation if either p−1 or p+1 does not have any large prime divisors. Similarly, an elliptic curve encryption or signature generation will be subject to an attack if the value of the private key d is very small as compared to the size of the underlying finite field. See ANSI Standard X9.3 1-1998 Digital Signatures Using Reversible Public Key Cryptography For the Financial Services Industry (rDSA) and ANSI Standard X9.62-1998 The Elliptic Curve Digital Signature Algorithm for the details of these methods.
While the algorithm specified in this invention creates “strong” primes in the definitions of the ANSI Standard X9.31 (that is, p−1, p+1, q−1, and q+1 all have large prime factors and |p−q| is large), the present invention should alleviate a requirement that such be present since the probability of such occurrence at random is extremely low and by not allowing many repeated efforts to generate a pair of primes, imposing such restrictions becomes unnecessary.